# Difference between revisions of "Euler problems/51 to 60"

(→[http://projecteuler.net/index.php?section=problems&id=55 Problem 55]: a solution) |
(→[http://projecteuler.net/index.php?section=problems&id=56 Problem 56]: a solution) |
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Line 44: | Line 44: | ||

== [http://projecteuler.net/index.php?section=problems&id=56 Problem 56] == |
== [http://projecteuler.net/index.php?section=problems&id=56 Problem 56] == |
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− | Considering natural numbers of the form, |
+ | Considering natural numbers of the form, a<sup>b</sup>, finding the maximum digital sum. |

Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | problem_56 = undefined |
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+ | problem_56 = maximum [dsum (a^b) | a <- [1..99], b <-[1..99]] |
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+ | where dsum 0 = 0 |
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+ | dsum n = let ( d, m ) = n `divMod` 10 in m + ( dsum d ) |
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</haskell> |
</haskell> |
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## Revision as of 07:24, 30 March 2007

## Contents

## Problem 51

Find the smallest prime which, by changing the same part of the number, can form eight different primes.

Solution:

```
problem_51 = undefined
```

## Problem 52

Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits in some order.

Solution:

```
problem_52 = undefined
```

## Problem 53

How many values of C(n,r), for 1 ≤ n ≤ 100, exceed one-million?

Solution:

```
problem_53 = undefined
```

## Problem 54

How many hands did player one win in the game of poker?

Solution:

```
problem_54 = undefined
```

## Problem 55

How many Lychrel numbers are there below ten-thousand?

Solution:

```
problem_55 = length $ filter isLychrel [1..9999]
where isLychrel n = all notPalindrome (take 50 (tail (iterate revadd n)))
notPalindrome s = (show s) /= reverse (show s)
revadd n = n + rev n
where rev n = read (reverse (show n))
```

## Problem 56

Considering natural numbers of the form, a^{b}, finding the maximum digital sum.

Solution:

```
problem_56 = maximum [dsum (a^b) | a <- [1..99], b <-[1..99]]
where dsum 0 = 0
dsum n = let ( d, m ) = n `divMod` 10 in m + ( dsum d )
```

## Problem 57

Investigate the expansion of the continued fraction for the square root of two.

Solution:

```
problem_57 = undefined
```

## Problem 58

Investigate the number of primes that lie on the diagonals of the spiral grid.

Solution:

```
problem_58 = undefined
```

## Problem 59

Using a brute force attack, can you decrypt the cipher using XOR encryption?

Solution:

```
problem_59 = undefined
```

## Problem 60

Find a set of five primes for which any two primes concatenate to produce another prime.

Solution:

```
problem_60 = undefined
```